Based on the mean-field approximation and the phase space analysis, we study the dynamics of an atom-molecule conversion system subject to particle loss. Starting from the many-body dynamics described by a master equation, an effective nonlinear Schrödinger equation is introduced. The classical phase space is then specified and classified by fixed points. The boundary, which separate different dynamical regimes have been calculated and discussed. The effect of particle loss on the conversion efficiency and the self-trapping is explored.

Beyond the quantum Markov approximation and the weak coupling limit, we present a general theory to calculate the geometric phase for open systems with and without conserved energy. As an example, the geometric phase for a two-level system coupling both dephasingly and dissipatively to its environment is calculated. Comparison with the results from quantum trajectory analysis is presented and discussed.

Beyond the quantum Markov approximation, we calculate the geometric phase of a two-level system driven by a quantized magnetic field subject to phase dephasing. The phase reduces to the standard geometric phase in the weak coupling limit and it involves the phase information of the environment in general. In contrast with the geometric phase in dissipative systems, the geometric phase acquired by the system can be observed on a long time scale. We also show that with the system decohering to its pointer states, the geometric phase factor tends to a sum over the phase factors pertaining to the pointer states.PACS numbers: 03.65. Vf, 03.65.Yz Quantum mechanics tell us that physical states are equivalent up to a global phase, which in general does not contain useful information about the described system and thus can be ignored. This is not the case, however, for a system transported round a circuit by varying the parameters s = (s 1 , s 2 , ...) in its Hamiltonian H( s). As Berry showed [1], the phase can have a component of geometric origin called geometric phase with important observable consequences, such as the Aharonov-Bohm effect [2] and the spin-1 2 particle driven by a rotating magnetic field [1]. The geometric phases that only depend on the path followed by the system during its evolution, have been investigated and tested in a variety of settings and have been generalized in several directions [3]. The geometric phases are attractive both from a theoretical perspective, and from the point of view of possible applications, among which geometric quantum computation [4,5,6,7] is one of the most importance.As realistic systems always interact with their environment, the study on the geometric phase in open systems become interesting. Garrison and Wright [8] were the first to touch on this issue by describing open system evolution in terms of a non-Hermitian Hamiltonian. This is a pure state analysis, so it did not address the problem of geometric phases for mixed states. Toward the geometric phase for mixed states in open systems, the approaches used involve solving the master equation of the system [9,10,11,12,13], employing a quantum trajectory analysis [14,15] or Krauss operators [16], and the perturbative expansions [17,18]. Some interesting results were achieved, briefly summarized as follows: nonhermitian Hamiltonian lead to a modification of Berry's phase [8,17], stochastically evolving magnetic fields produce both energy shift and broadening [18], phenomenological weakly dissipative Liouvillians alter Berry's phase by introducing an imaginary correction [11] or lead to damping and mixing of the density matrix elements [12]. However, almost all these studies are performed for dissipative systems, and thus the representations are applicable for systems whose energy is not conserved. For open systems with conserved energy (dephasing systems), the problem beyond the Markov approximation remains untouched to our best knowledge. Because the systemenvironment interaction H I and the free system Hamilton...

We explore the quantum version of the Brayton cycle with a composite system as the working substance. The actual Brayton cycle consists of two adiabatic and two isobaric processes. Two pressures can be defined in our isobaric process; one corresponds to the external magnetic field (characterized by F(x)) exerted on the system, while the other corresponds to the coupling constant between the subsystems (characterized by F(y)). As a consequence, we can define two types of quantum Brayton cycle for the composite system. We find that the subsystem experiences a quantum Brayton cycle in one quantum Brayton cycle (characterized by F(x)), whereas the subsystem's cycle is quantum Otto cycle in another Brayton cycle (characterized by F(y)). The efficiency for the composite system equals to that for the subsystem in both cases, but the work done by the total system is usually larger than the sum of the work done by the two subsystems. The other interesting finding is that for the cycle characterized by F(y), the subsystem can be a refrigerator, while the total system is a heat engine. The result in this paper can be generalized to a quantum Brayton cycle with a general coupled system as the working substance.